DOCSLIB.ORG

15. Bayesian Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
15. Bayesian Methods 15. Bayesian Methods c A. Colin Cameron & Pravin K. Trivedi 2006 These transparencies were prepared in 2003. They can be used as an adjunct to Chapter 13 of our subsequent book Microeconometrics: Methods and Applications Cambridge University Press, 2005. Original version of slides: May 2003 Outline 1. Introduction 2. Bayesian Approach 3. Bayesian Analysis of Linear Regression 4. Monte Carlo Integration 5. Markov Chain Monte Carlo Simulation 6. MCMC Example: Gibbs Sampler for SUR 7. Data Augmentation 8. Bayesian Model Selection 9. Practical Considerations 1 Introduction Bayesian regression has grown greatly since books by Arnold Zellner (1971) and Leamer (1978). Controversial. Requires specifying a probabilistic model of prior beliefs about the unknown parameters. [Though role of prior is negligible in large samples and relatively uninformative priors can be speci…ed.] Growth due to computational advances. In particular, despite analytically intractable poste- rior can use simulation (Monte Carlo) methods to – estimate posterior moments – make draws from the posterior. 2 Bayesian Approach 1. Prior () Uncertainty about parameters explicitly modelled by density (). e.g. is an income elasticity and on basis of eco- nomic model or previous studies it is felt that Pr[0:8 1:2] = 0:95. Possible prior is [1; 0:12]. N 2. Sample joint density or likelihood f(y ) j Similar to ML framework. In single equation case y is N 1 vector and depen- dence on regressors X is suppressed. 3. Posterior p( y) j Obtained by combining prior and sample. 2.1 Bayes Theorem Bayes inverse law of probability gives posterior f(y )() p( y) = j ; (1) j f(y) where f(y) is marginal (wrt ) prob. distn of y f(y) = f(y )()d: (2) Z j Pr[A B] Pr[B A] Pr[A] Proof: Use Pr[A B] = \ = j : j Pr[B] Pr[B] f(y) in (1) is free of , so can write p( y) as j proportional to the product of the pdf and the prior p( y) f(y )(): (3) j _ j Big di¤erence – Frequentist: 0 is constant and is random. – Bayesian: is random. b 2.2 Normal-Normal iid Example 1. Sample Density f(y ) j Assume y [; 2] with unknown and 2 ij N given. N N=2 f(y ) = 22 exp (y )2 =22 j 8 i 9 < iX=1 = N exp (y :)2 ; ; _ 2 2 2. Prior (). Suppose [; 2] where and 2 are given. N 1=2 () = 2 2 exp ( )2 =2 2 1 n o exp ( )2 ; _ 2 2 3. Posterior density p( y) j N 1 p( y) exp (y )2 exp ( )2 _ 2 2 j 2 2 After some algebra (completing the square) 1 ( )2 (y )2 p( y) exp + _ 2 12 2 j (2 " N + #) 2 1 ( 1) _ exp 2 (2 " 1 #) 2 2 2 1 = 1 Ny= + = 2 2 2 1 1 = N= + 1= : Properties of posterior: – Posterior density is y [ ; 2]: j N 1 1 – Posterior mean 1 is weighted average of prior mean and sample average y. 2 – Posterior precision 1 is sum of sample preci- sion of y, N=2, and prior precision 1= 2. [Precision is the reciprocal of the variance.] – As N , y [y; 2=N] ! 1 j N Normal-Normal Example with 2 = 100; = 5, 2 = 3, N = 50 and y = 10. 2.3 Speci…cation of the Prior Tricky. Not the focus of this talk. Prior can be improper yet yield proper posterior. A noninformative prior has little impact on the re- sulting posterior distribution. Use Je¤reys prior, not uniform prior, as invariant to reparametrization. For informative prior prefer natural conjugate prior as it yields analytical posterior. Exponential family prior, density and posterior. ) e.g. normal-normal, Poisson-gamma Hierarchical priors popular for multilevel models. 2.4 Measures Related to Posterior Marginal Posterior: p(k y) = p(1; :::; d y)d1::dk 1dk+1::dd: j j R Posterior Moments: mean/median; standard devn. Point Estimation: no unknown to estimate. 0 Instead …nd value of that minimizes a loss function. Posterior Intervals (95%): Pr ) y = 0:95. k;:025 k k;:975 j h i Hypothesis Testing: Not relevant. Bayes factors. Conditional Posterior Density: p(k j; j k; y) = p( y)=p(j k y): j 2 j 2 j 2.5 Large Sample Behavior of Posterior Asymptotically role of prior disappears. If there is a true then the posterior mode (max- 0 imum of the posterior) is consistent for this. b Posterior is asymptotically normal a y ; () 1 ; (4) j N I h i centered around the posteriorb mode,b where 1 @2 ln p( y) = j : I " @@ # 0 = b b Called a Bayesian central limit theorem. 3 Bayesian Linear Regression Linear regression model y X; ;2 [X ;2I ]: j N N Di¤erent results with noninformative and informative priors. Even within these get di¤erent results according to setup. 3.1 Noninformative Priors Je¤reys’priors: ( ) c and (2) 1=2. j _ _ All values of j equally likely. Smaller values of 2 are viewed as more likely. 2 2 ( ; ) _ 1= : Posterior density after some algebra p( ;2 y; X) j 1 K=2 1 1 exp ( ) (X X)( ) _ 2 0 2 0 2 1 (N K)=2+1 b (N K) s2 b exp 2 2 2 ! 2 2 1 Cond. posterior p( ; y; X) is [ ; X X ] j N OLS 0 b Marginal posterior p( y; X) (integrate out 2) is j multivariate t-distribution centered at with N K 2 1 dof and variance s (N K) X X = (N K 2). 0 b Marginal posterior p(2 y; X) is inverse gamma. j Qualitatively similar to frequentist analysis in …nite samples. Interpretation is quite di¤erent. E.g. Bayesian 95 percent posterior interval for j is j t:025;N K se j h i b b – means that j lies in this interval with posterior probability 0.95 – not that if we had many samples and constructed many such intervals 95 percent of them will con- tain the true j0. 3.2 Informative Priors Use conjugate priors: – Prior for 2 is [ ; 2 ]: j N 0 0 – Prior for 2 is inverse-gamma. Posterior after much algebra is p( ;1= y; X) j (0+N)=2 1 s1 K=2 2 exp 2 _ 2 2 1 exp 0 22 1 where 1 = 0 + X0X ( 0 0 + X0X ) 1 = 0 + X0X b 1 1 s = s + u u + 0 + X X 1 0 0 0 0 b b Conditional posterior p( 2; y; X) is [ ; ]: j N 1 Marginal posterior p( y; X) (integrate out 2) is j multivariate t-distribution centered at : Here is average of and prior mean : OLS 0 And precision is sum of prior and sample precisions. b 4 Monte Carlo Integration Compute key posterior moments, without …rst ob- taining the posterior distribution. Want E[m( y)], where expectation is wrt to pos- j terior density p( y). j For notational convenience suppress y: So wish to compute E [m()] = m()p()d: (5) Z Need a numerical estimate of an integral: - Numerical quadrature too hard. - Direct Monte Carlo with draws from p() not pos- sible. - Instead use importance sampling. 4.1 Importance Sampling Rewrite E [m()] = m()p()d Z m()p() = g()d; Z g() ! where g() > 0 is a known density with same sup- port as p(). The corresponding Monte Carlo integral estimate is 1 S m(s)p(s) E [m()] = ; (6) S g(s) sX=1 where s,bs = 1; :::; S, are S draws from of from g() not p(). To apply to posterior need also to account for con- stant of integration in the denominator of (1). Let pker() = f (y ) () be posterior kernel. j Then posterior density is pker() p() = ; pker()d with posterior moment R pker() E [m()] = m() d ker Z p ()d! m() pker()d = R ker R p ()d ker mR () p ()=g() g()d = : R pker()=g() g()d R The importance sampling-based estimate is then 1 S m(s)pker(s)=g(s) E [m()] = S s=1 ; (7) 1 S ker s s PS s=1 p ( )=g( ) b where s, s = 1; :::;P S, are S draws of from the importance sampling density g(): Method was proposed by Kloek and van Dijk (1978). Geweke (1989) established consistency and asymp- totic normality as S if ! 1 – E[m()] < so the posterior moment exists 1 – p()d = 1 so the posterior density is proper. May require ()d < . R 1 R – g() > 0 over the support of p() – g() should have thicker tails than the p() to ensure that the importance weight w() = p()=g() remains bounded. e.g. use multivariate- t. The importance sampling method can be used to estimates many quantities, including mean, standard deviation and percentiles of the posterior. 5 Markov Chain Monte Carlo Sim- ulation If can make S draws from the posterior, 1 s E[m()] can be estimated by S s m( ). P But hard to make draws if no tractable closed form expression for the posterior density. Instead make sequential draws that, if the sequence is run long enough, converge to a stationary distrib- ution that coincides with the posterior density p(). Called Markov chain Monte Carlo, as it involves simulation (Monte Carlo) and the sequence is that of a Markov chain. Note that draws are correlated. 5.1 Markov Chains A Markov chain is a sequence of random variables xn (n = 0; 1; 2; :::) with Pr [xn+1 = x xn; xn 1; :::; x0] = Pr [xn+1 = x xn] ; j j so that the distribution of xn+1 given past is com- pletely determined only by the preceding value xn.
Load more

Recommended publications
  • The Exponential Family 1 Definition
    The Exponential Family David M. Blei Columbia University November 9, 2016 The exponential family is a class of densities (Brown, 1986). It encompasses many familiar forms of likelihoods, such as the Gaussian, Poisson, multinomial, and Bernoulli. It also encompasses their conjugate priors, such as the Gamma, Dirichlet, and beta. 1 Definition A probability density in the exponential family has this form p.x / h.x/ exp >t.x/ a./ ; (1) j D f g where is the natural parameter; t.x/ are sufficient statistics; h.x/ is the “base measure;” a./ is the log normalizer. Examples of exponential family distributions include Gaussian, gamma, Poisson, Bernoulli, multinomial, Markov models. Examples of distributions that are not in this family include student-t, mixtures, and hidden Markov models. (We are considering these families as distributions of data. The latent variables are implicitly marginalized out.) The statistic t.x/ is called sufficient because the probability as a function of only depends on x through t.x/. The exponential family has fundamental connections to the world of graphical models (Wainwright and Jordan, 2008). For our purposes, we’ll use exponential 1 families as components in directed graphical models, e.g., in the mixtures of Gaussians. The log normalizer ensures that the density integrates to 1, Z a./ log h.x/ exp >t.x/ d.x/ (2) D f g This is the negative logarithm of the normalizing constant. The function h.x/ can be a source of confusion. One way to interpret h.x/ is the (unnormalized) distribution of x when 0. It might involve statistics of x that D are not in t.x/, i.e., that do not vary with the natural parameter.
    [Show full text]
  • 1 Introduction to Bayesian Inference 2 Introduction to Gibbs Sampling
    MCMC I: July 5, 2016 1 MCMC I 8th Summer Institute in Statistics and Modeling in Infectious Diseases Course Time Plan July 13-15, 2016 Instructors: Vladimir Minin, Kari Auranen, M. Elizabeth Halloran Course Description: This module is an introduction to Markov chain Monte Carlo methods with some simple applications in infectious disease studies. The course includes an introduction to Bayesian inference, Monte Carlo, MCMC, some background theory, and convergence diagnostics. Algorithms include Gibbs sampling and Metropolis-Hastings and combinations. Programming is in R. Familiarity with the R statistical package or other computing language is needed. Course schedule: The course is composed of 10 90-minute sessions, for a total of 15 hours of instruction. 1 Introduction to Bayesian Inference • Overview of the course. • Bayesian inference: Likelihood, prior, posterior, normalizing constant • Conjugate priors; Beta-binomial; Poisson-gamma; normal-normal • Posterior summaries, mean, mode, posterior intervals • Motivating examples: Chain binomial model (Reed-Frost), General Epidemic Model, SIS model. • Lab: – Goals: Warm-up with R for simple Bayesian computation – Example: Posterior distribution of transmission probability with a binomial sampling distribution using a conjugate beta prior distribution – Summarizing posterior inference (mean, median, posterior quantiles and intervals) – Varying the amount of prior information – Writing an R function 2 Introduction to Gibbs Sampling • Chain binomial model and data augmentation • Brief introduction
    [Show full text]
  • Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation
    Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation Thomas Brouwer Jes Frellsen Pietro Lio’ Computer Laboratory Department of Engineering Computer Laboratory University of Cambridge University of Cambridge University of Cambridge United Kingdom United Kingdom United Kingdom [email protected] [email protected] [email protected] Abstract We present a fast variational Bayesian algorithm for performing non-negative matrix factorisation and tri-factorisation. We show that our approach achieves faster convergence per iteration and timestep (wall-clock) than Gibbs sampling and non-probabilistic approaches, and do not require additional samples to estimate the posterior. We show that in particular for matrix tri-factorisation convergence is difficult, but our variational Bayesian approach offers a fast solution, allowing the tri-factorisation approach to be used more effectively. 1 Introduction Non-negative matrix factorisation methods Lee and Seung [1999] have been used extensively in recent years to decompose matrices into latent factors, helping us reveal hidden structure and predict missing values. In particular we decompose a given matrix into two smaller matrices so that their product approximates the original one. The non-negativity constraint makes the resulting matrices easier to interpret, and is often inherent to the problem – such as in image processing or bioinformatics (Lee and Seung [1999], Wang et al. [2013]). Some approaches approximate a maximum likelihood (ML) or maximum a posteriori (MAP) solution that minimises the difference between the observed matrix and the decomposition of this matrix. This gives a single point estimate, which can lead to overfitting more easily and neglects uncertainty. Instead, we may wish to find a full distribution over the matrices using a Bayesian approach, where we define prior distributions over the matrices and then compute their posterior after observing the actual data.
    [Show full text]
  • Bayesian Inference in the Normal Linear Regression Model
    Bayesian Inference in the Normal Linear Regression Model () Bayesian Methods for Regression 1 / 53 Bayesian Analysis of the Normal Linear Regression Model Now see how general Bayesian theory of overview lecture works in familiar regression model Reading: textbook chapters 2, 3 and 6 Chapter 2 presents theory for simple regression model (no matrix algebra) Chapter 3 does multiple regression In lecture, I will go straight to multiple regression Begin with regression model under classical assumptions (independent errors, homoskedasticity, etc.) Chapter 6 frees up classical assumptions in several ways Lecture will cover one way: Bayesian treatment of a particular type of heteroskedasticity () Bayesian Methods for Regression 2 / 53 The Regression Model Assume k explanatory variables, xi1,..,xik for i = 1, .., N and regression model: yi = b1 + b2xi2 + ... + bk xik + #i . Note xi1 is implicitly set to 1 to allow for an intercept. Matrix notation: y1 y2 y = 2 . 3 6 . 7 6 7 6 y 7 6 N 7 4 5 # is N 1 vector stacked in same way as y () Bayesian Methods for Regression 3 / 53 b is k 1 vector X is N k matrix 1 x12 .. x1k 1 x22 .. x2k X = 2 ..... 3 6 ..... 7 6 7 6 1 x .. x 7 6 N2 Nk 7 4 5 Regression model can be written as: y = X b + #. () Bayesian Methods for Regression 4 / 53 The Likelihood Function Likelihood can be derived under the classical assumptions: 1 2 # is N(0N , h IN ) where h = s . All elements of X are either fixed (i.e. not random variables). Exercise 10.1, Bayesian Econometric Methods shows that likelihood function can be written in
    [Show full text]
  • An Introduction to Markov Chain Monte Carlo Methods and Their Actuarial Applications
    AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS AND THEIR ACTUARIAL APPLICATIONS DAVID P. M. SCOLLNIK Department of Mathematics and Statistics University of Calgary Abstract This paper introduces the readers of the Proceed- ings to an important class of computer based simula- tion techniques known as Markov chain Monte Carlo (MCMC) methods. General properties characterizing these methods will be discussed, but the main empha- sis will be placed on one MCMC method known as the Gibbs sampler. The Gibbs sampler permits one to simu- late realizations from complicated stochastic models in high dimensions by making use of the model’s associated full conditional distributions, which will generally have a much simpler and more manageable form. In its most extreme version, the Gibbs sampler reduces the analy- sis of a complicated multivariate stochastic model to the consideration of that model’s associated univariate full conditional distributions. In this paper, the Gibbs sampler will be illustrated with four examples. The first three of these examples serve as rather elementary yet instructive applications of the Gibbs sampler. The fourth example describes a reasonably sophisticated application of the Gibbs sam- pler in the important arena of credibility for classifica- tion ratemaking via hierarchical models, and involves the Bayesian prediction of frequency counts in workers compensation insurance. 114 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 115 1. INTRODUCTION The purpose of this paper is to acquaint the readership of the Proceedings with a class of simulation techniques known as Markov chain Monte Carlo (MCMC) methods. These methods permit a practitioner to simulate a dependent sequence of ran- dom draws from very complicated stochastic models.
    [Show full text]
  • Accelerating Bayesian Inference on Structured Graphs Using Parallel Gibbs Sampling
    Accelerating Bayesian Inference on Structured Graphs Using Parallel Gibbs Sampling Glenn G. Ko∗, Yuji Chai∗, Rob A. Rutenbary, David Brooks∗, Gu-Yeon Wei∗ ∗Harvard University, Cambridge, USA yUniversity of Pittsburgh, Pittsburgh, USA [email protected], [email protected], [email protected], fdbrooks, [email protected] Abstract—Bayesian models and inference is a class of machine with substantial uncertainty and noise is normally reserved for learning that is useful for solving problems where the amount Bayesian models and inference. of data is scarce and prior knowledge about the application Bayesian models pose challenges that are different from allows you to draw better conclusions. However, Bayesian models often requires computing high-dimensional integrals and finding deep learning. They tend to have more complex structure, the posterior distribution can be intractable. One of the most that may be hierarchical with dependencies between different commonly used approximate methods for Bayesian inference distributions Without an easily-defined common computational is Gibbs sampling, which is a Markov chain Monte Carlo kernel and hardware accelerators like GPU and TPU [5], and (MCMC) technique to estimate target stationary distribution. software frameworks such as Tensorflow [6] and PyTorch [7], The idea in Gibbs sampling is to generate posterior samples by iterating through each of the variables to sample from its it is hard to build and run models fast and efficiently. This conditional given all the other variables fixed. While Gibbs work explores hardware accelerators for Bayesian models and sampling is a popular method for probabilistic graphical models inference which has growing interest and demand. such as Markov Random Field (MRF), the plain algorithm is Probabilistic graphical models are Bayesian models described slow as it goes through each of the variables sequentially.
    [Show full text]
  • Gibbs Sampling, Exponential Families and Orthogonal Polynomials
    Statistical Science 2008, Vol. 23, No. 2, 151–178 DOI: 10.1214/07-STS252 c Institute of Mathematical Statistics, 2008 Gibbs Sampling, Exponential Families and Orthogonal Polynomials1 Persi Diaconis, Kshitij Khare and Laurent Saloff-Coste Abstract. We give families of examples where sharp rates of conver- gence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, expo- nential families, conjugate priors, location families, orthogonal polyno- mials, singular value decomposition. 1. INTRODUCTION which has f as stationary density under mild con- ditions discussed in [4, 102]. The Gibbs sampler, also known as Glauber dy- The algorithm was introduced in 1963 by Glauber namics or the heat-bath algorithm, is a mainstay of [49] to do simulations for Ising models, and inde- scientific computing. It provides a way to draw sam- pendently by Turcin [103]. It is still a standard tool ples from a multivariate probability density f(x1,x2, of statistical physics, both for practical simulation ...,xp), perhaps only known up to a normalizing (e.g., [88]) and as a natural dynamics (e.g., [10]). constant, by a sequence of one-dimensional sampling The basic Dobrushin uniqueness theorem showing problems. From (X1,...,Xp) proceed to (X1′ , X2,..., existence of Gibbs measures was proved based on Xp), then (X1′ , X2′ , X3,...,Xp),..., (X1′ , X2′ ,...,Xp′ ) this dynamics (e.g., [54]). It was introduced as a where at the ith stage, the coordinate is sampled base for image analysis by Geman and Geman [46].
    [Show full text]
  • Gibbs Sampling
    4 Modern Model Estimation Part 1: Gibbs Sampling The estimation of a Bayesian model is the most difficult part of undertaking a Bayesian analysis. Given that researchers may use different priors for any particular model, estimation must be tailored to the specific model under consideration. Classical analyses, on the other hand, often involve the use of standard likelihood functions, and hence, once an estimation routine is developed, it can be used again and again. The trade-off for the additional work required for a Bayesian analysis is that (1) a more appropriate model for the data can be constructed than extant software may allow, (2) more measures of model fit and outlier/influential case diagnostics can be produced, and (3) more information is generally available to summarize knowledge about model parameters than a classical analysis based on maximum likelihood (ML) estimation provides. Along these same lines, additional measures may be constructed to test hypotheses concerning parameters not directly estimated in the model. In this chapter, I first discuss the goal of model estimation in the Bayesian paradigm and contrast it with that of maximum likelihood estimation. Then, I discuss modern simulation/sampling methods used by Bayesian statisticians to perform analyses, including Gibbs sampling. In the next chapter, I discuss the Metropolis-Hastings algorithm as an alternative to Gibbs sampling. 4.1 What Bayesians want and why As the discussion of ML estimation in Chapter 2 showed, the ML approach finds the parameter values that maximize the likelihood function for the ob- served data and then produces point estimates of the standard errors of these estimates.
    [Show full text]
  • Differentially Private Bayesian Inference for Exponential Families
    Differentially Private Bayesian Inference for Exponential Families Garrett Bernstein College of Information and Computer Sciences University of Massachusetts Amherst Amherst, MA 01002 [email protected] Daniel Sheldon College of Information and Computer Sciences University of Massachusetts Amherst Amherst, MA 01002 [email protected] Abstract The study of private inference has been sparked by growing concern regarding the analysis of data when it stems from sensitive sources. We present the first method for private Bayesian inference in exponential families that properly accounts for noise introduced by the privacy mechanism. It is efficient because it works only with sufficient statistics and not individual data. Unlike other methods, it gives properly calibrated posterior beliefs in the non-asymptotic data regime. 1 Introduction Differential privacy is the dominant standard for privacy [1]. A randomized algorithm that satisfies differential privacy offers protection to individuals by guaranteeing that its output is insensitive to changes caused by the data of any single individual entering or leaving the data set. An algorithm can be made differentially private by applying one of several general-purpose mechanisms to randomize the computation in an appropriate way, for example, by adding noise calibrated to the sensitivity of the quantity being computed, where sensitivity captures how much the quantity depends on any individual’s data [1]. Due to the obvious importance of protecting individual privacy while drawing population level inferences from data, differentially private algorithms have been developed for a broad range of machine learning tasks [2–9]. There is a growing interest in private methods for Bayesian inference [10–14].
    [Show full text]
  • A New Hyperprior Distribution for Bayesian Regression Model with Application in Genomics
    bioRxiv preprint doi: https://doi.org/10.1101/102244; this version posted January 22, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. A New Hyperprior Distribution for Bayesian Regression Model with Application in Genomics Renato Rodrigues Silva 1,* 1 Institute of Mathematics and Statistics, Federal University of Goi´as, Goi^ania,Goi´as,Brazil *[email protected] Abstract In the regression analysis, there are situations where the model have more predictor variables than observations of dependent variable, resulting in the problem known as \large p small n". In the last fifteen years, this problem has been received a lot of attention, specially in the genome-wide context. Here we purposed the bayes H model, a bayesian regression model using mixture of two scaled inverse chi square as hyperprior distribution of variance for each regression coefficient. This model is implemented in the R package BayesH. Introduction 1 In the regression analysis, there are situations where the model have more predictor 2 variables than observations of dependent variable, resulting in the problem known as 3 \large p small n" [1]. 4 To figure out this problem, there are already exists some methods developed as ridge 5 regression [2], least absolute shrinkage and selection operator (LASSO) regression [3], 6 bridge regression [4], smoothly clipped absolute deviation (SCAD) regression [5] and 7 others. This class of regression models is known in the literature as regression model 8 with penalized likelihood [6]. In the bayesian paradigma, there are also some methods 9 purposed as stochastic search variable selection [7], and Bayesian LASSO [8].
    [Show full text]
  • 9 Introduction to Hierarchical Models
    9 Introduction to Hierarchical Models One of the important features of a Bayesian approach is the relative ease with which hierarchical models can be constructed and estimated using Gibbs sampling. In fact, one of the key reasons for the recent growth in the use of Bayesian methods in the social sciences is that the use of hierarchical models has also increased dramatically in the last two decades. Hierarchical models serve two purposes. One purpose is methodological; the other is substantive. Methodologically, when units of analysis are drawn from clusters within a population (communities, neighborhoods, city blocks, etc.), they can no longer be considered independent. Individuals who come from the same cluster will be more similar to each other than they will be to individuals from other clusters. Therefore, unobserved variables may in- duce statistical dependence between observations within clusters that may be uncaptured by covariates within the model, violating a key assumption of maximum likelihood estimation as it is typically conducted when indepen- dence of errors is assumed. Recall that a likelihood function, when observations are independent, is simply the product of the density functions for each ob- servation taken over all the observations. However, when independence does not hold, we cannot construct the likelihood as simply. Thus, one reason for constructing hierarchical models is to compensate for the biases—largely in the standard errors—that are introduced when the independence assumption is violated. See Ezell, Land, and Cohen (2003) for a thorough review of the approaches that have been used to correct standard errors in hazard model- ing applications with repeated events, one class of models in which repeated measurement yields hierarchical clustering.
    [Show full text]
  • Chapter 4 MONTE CARLO METHODS and MCMC SIMULATION
    Chapter 4 MONTE CARLO METHODS AND MCMC SIMULATION In this chapter we treat the theory required for learning about pos- terior distributions, and we lay the foundation needed in order to learn BNs from incomplete data. We also develop a Markov chain Monte Carlo sampler called MB-MCMC for obtaining DAG models from the posterior model distribution given complete data. Monte Carlo methods is a broad topic, and in this chapter we only present the essentials. See for instance Robert and Casella, 2002; Gilks et al., 1996 for a thorough treatment of different Monte Carlo tech- niques, including Markov chain Monte Carlo simulation and practical applications. 1. Monte Carlo methods We investigate the generic problem of computing the following sum- mation (replace sums with integrals for continuous variables): EPr[h(X)] = h(x)Pr(x) (4.1) Xx In particular, h(X) can be a probability distribution, for instance the conditional distribution Pr(Y |X), in which case the expectation coin- cides with the distribution Pr(Y ). If eq. 4.1 is difficult to solve analytically or is infeasible to compute because of an extremely large cardinality of ΩX, it can be approximated. Sample X(1),..., X(n) from Pr(X), i.e., X(t) ∼ Pr(X), and compute 38 EFFICIENT LEARNING OF BAYESIAN NETWORKS the empirical average of eq. 4.1: n 1 E [h(X)] ≈ h(x(t)) (4.2) Pr n Xt=1 The Law of Large Numbers guarantees that the approximation converges towards the expectation in eq. 4.1 as n → ∞. For one reason or another it might be the case that sampling from Pr(X) is undesirable.
    [Show full text]